Problems.

1. A ball is thrown vertically upward with a velocity of 64 ft per sec.; how soon will it reach the ground again and how high will it rise, and what will be its velocity when half-way up?

2. A falling body has a velocity 200 cm. sec.; how far will it drop before its velocity becomes 10,000 cm. sec.? Take g 980.

3. A weight thrown forward on ice with velocity 60 ft. per sec. is resisted by a constant force, and after 5 seconds has half its original velocity; how far has it gone in that time?

4. Find the acceleration in the previous problem, also how far the weight will go before coming to rest.

5. A mass of 10 grms. is acted on by a constant force which changes its velocity from 100 to 500 cms. per sec. in 5 seconds. Find the acceleration and amount of the force.

6. What steady forward pull must be exerted by a locomotive in starting a 200-ton train to give it a velocity of 20 miles per hour in 5 minutes, neglecting friction. Find force in poundals and then in pounds.

7. A weight of ten pounds is thrown forward on ice with a velocity 50; if the coefficient of friction between it and the ice is 0.10, how far will it go and in how many seconds will it stop.

8. A 300-lb. mass is lowered by a rope with uniform velocity. What is the tension on the rope? If it is lowered with a constant acceleration of 10 ft. per sec. per sec. what is the tension? What if it is lowered with acceleration g?

9. If a man weighing 75 kilograms is in an elevator which is going up with constant velocity, how much force does he exert on its floor? What if the elevator has an upward acceleration of 3 meter-sec.?

10. What is the least acceleration with which a man weighing 150 lbs. can slide down a fire-escape rope which can only sustain a weight of 100 lbs.? And what velocity will the man have after sliding 50 ft.? 11. A 30-grm. weight is drawn up by a 70-grm. weight by means of a cord over a frictionless pulley. Find the acceleration (taking g=980) and also the tension on the cord. How far will the weights move in 3 seconds from the start?

12. A 38-lb. weight resting on a level, frictionless table is drawn along by a 4-lb. weight by means of a cord over a frictionless pulley. Find the acceleration and also the tension on the cord.

13. If in the previous problem the friction between the weight and table is a force of 2 pounds, find acceleration and tension as before.

14. How many foot-poundals of work are required to give a 500-lb. shell a velocity of 2000 ft. per second? Find the work also in footpounds. If this work is done by the powder gas in a gun 25 ft. long, find the average force in pounds against the shell as it is discharged.

15. How much energy in foot-pounds must be expended in giving a 300-ton train a velocity of 30 miles an hour? If the locomotive works at the rate of 100 H. P., how long will it take to bring the train up to speed?

16. A 3-kilogram hammer with a velocity of 5 meters per sec. drives a nail 4 cm. into a plank. Find the average resistance in dynes and grams and how much weight resting on the nail would be required to force it into the wood.

17. A bullet weighing one ounce and having a velocity of 1000 ft. sec. is fired through a plank 3 inches thick which resists it with a force of 800 pounds. With what velocity will it come out, and how many such planks could it pierce?

18. A bullet weighing 15 grms. is shot into a suspended block of wood weighing 2985 grms. and gives it a velocity of 200 cms. per sec.; find the velocity of the bullet.

19. How high above its original level will the suspended block, in the last question, swing in consequence of the velocity given to it? 20. If all the energy of a 640-lb. shell having a velocity of 2000 ft. sec. could be spent in raising a 10,000-ton battle ship, how high would it lift it?

21. A bullet weighing 10 grams has a velocity of 600 meters

per second and penetrates 30 cms. into a pine log. What is the force in kilograms with which the bullet is resisted, and how far would it penetrate if it had half the original velocity?

22. A monkey clings to one end of a rope passing over a frictionless pulley, and is balanced by an exactly equal weight on the other end of the rope. Explain what will happen to the counterpoise if the monkey climbs ten ft. up the rope and then suddenly stops. The mass of the rope and wheel are to be neglected.

23. A cord passes over two fixed pulleys and hangs down vertically between them supporting a movable pulley which with attached weight weighs 5 lbs. A 3-lb. weight is hung on one end of the cord and a 4-lb. weight on the other end. Find the accelerations of all three weights and the tension on the cord.

MOTION OF A PARTICLE IN CURVED PATH.

110. Motion of a Projectile.-When a body near the surface of the earth is thrown in any direction, such as A B, it is subject to the steady force of the earth's attraction vertically downward, and therefore it has constantly the downward acceleration of gravity g. The initial impulse, however, gave it a forward velocity V in the direction AB,

B

in which direction it would have continued to move with constant velocity if no force had acted on it. The actual path in which it moves may then be regarded as the resultant of motion with constant velocity V in the direction AB combined with a motion downward with constant acceleration g. Thus after a time t the body will have traveled a distance AC Vt in the direction AB, but it will also have fallen from C to D a distance s=g t2.

a

h

E

H

d FIG. 51.-Curve of projectiles and jets.

If a is the angle of elevation of AB above the horizontal, and if d is the distance A E which the projectile has advanced in a horizontal direction and h is its height, we have

d = V t cos a

h = V t sin a-gt2.

The path traversed may be shown to be a parabola with its axis vertical and passing through the highest point of the path. The highest point is half-way between the point of projection and the point G where the projectile again reaches the earth. The distance A G is called the range, and is a maximum when the angle a is 45°.

These results are easily deduced from the above equations, but it must be borne in mind that the influence of air resistance has been neglected. This force in rapidly moving bodies, like bullets, may be very great and changes the form of the trajectory to something like that shown by the second curve from A to H. In consequence of this the maximum range in gunnery is found at a much smaller elevation.

The form of the path of a projectile or ball is beautifully shown by a water jet, for each particle in the jet is a freely falling body.

111. Curved Pitching. If a ball when thrown forward is rapidly rotated the resistance of the air causes it to swerve from

the path that it would otherwise take. This is seen in the curving of a pitched ball and in the drifting of projectiles from rifled guns. It results from the viscosity of air in consequence of which the rotating ball drags air in on one side and flings it out on the other as it advances.

Suppose, for example, that a ball is spinning about an axis perpendicular to the paper as shown by the curved arrow in figure 52, while it is moving forward in the direction of the

R

a

b

FIG. 52. Curving of pitched ball.

straight arrow c d; the rotation of the ball drags air in from the side at a and carries it around toward the front of the ball at c, giving it a greater forward momentum than is given to the air between c and b where the sur

face of the ball is spinning backward.

The force against the ball is therefore greater betweeen a and c than it is between b and c. Let P and p represent these pressures against the ball. Their resultant as shown in the diagram of forces is the oblique force R which in part resists the forward motion of the ball, but also has a component represented by F which is at right angles to the path of the ball and causes it to swerve to one side in the direction of the dotted line.

As the force F acts constantly it causes the ball to move sidewise with constantly accelerated motion, and therefore the curving rapidly increases as the ball advances.

112. Motion Around the Earth.-Suppose it were possible to shoot a cannon ball in a horizontal direction from the top of some high mountain on the earth with a velocity so great that while it advanced a mile it would drop just enough to follow the curvature of the earth. Then, if there were no air resistance, the ball would continue around the earth and return to its original point of projection with undiminished velocity and would therefore continue to circulate forever around the earth as a satellite.

For suppose A (Fig. 53) is the point from which the projectile is shot in the direction AB. As it advances it drops away from the line AB; but the earth's surface also drops away from AB

in consequence of its curvature, by about 8 inches in the first mile. If, therefore, the cannon ball has a velocity which will carry it a mile in the same time that it will drop 8 inches, it will, on reaching the end of the mile, say at C, be just as high above the earth as at the start and be moving in the direction CD, tangent at C. As the ball moves forward the force due to the earth's attraction is always at right angles to the direction of motion, and hence the speed of the ball is neither increased nor diminished and all the conditions of the motion remain constant.

The time required for a body to drop 8 inches toward the earth is found from the formula (§99)

s = 1 g t2.

Since 88 inches, or 0.66 ft., and taking g=32 ft. sec., we find t=0.20 sec.; hence our cannon ball must have a velocity of a mile in 0.2 sec. or 5 miles per second.

The complete calculation may be made thus. Let the ball drop a distance BC=s (Fig. 54) in going forward the distance AB=d. If R is the radius of the earth the relation between s and d may be found from the similarity of the triangles ACE and ADC from which we find